1) The catalog currently in use by PSE2018 is titled PSE16DB. I was expecting the catalog to be named PSE18.
That's a bit confusing, but PSE18 does not exist. It's the 16th version of Elements, but Adobe decided to give new names to its various softwares, based on years.
2) I no longer have a My Catalog 16 folder, which makes me wonder if the catalog folder currently in use is just my old PSE16 folder retitled.
You should have a 'My catalog' catalog folder created from your old catalog version. That catalog in the default location has a database file named 'catalog.pse16db'. When you installed PSE2018, you have probably agreed to convert your old catalog. In that case, the catalog (and catalog folder) keep the same catalog name. The original catalog of the previous PSE version is still there; it is renamed with a -1 suffix, and does work seamlessly with the previous pse version.
3) In the past I put my catalogs in a custom location, but I didn't do that this time. Then it is in the default location. Use the menu Help >> System Information to check.
How can I ensure that I have a properly converted PSE2018 catalog? Run Optimize? Run Repair? Download the trial version over again (if that's allowed) and start over?
You should see no difference in the Organizer, either the conversion fails (with error messages) or it is ok.
Note that if you don't trust the converted catalog, the catalog manager lets you convert it again. Just be sure to tick the checkbox to show 'already converted catalogs'.